Optimal. Leaf size=77 \[ -\frac {F^{-\frac {n \sqrt {1-a x}}{\sqrt {a x+1}}} \left (F^{\frac {\sqrt {1-a x}}{\sqrt {a x+1}}}\right )^n \text {Ei}\left (\frac {n \sqrt {1-a x} \log (F)}{\sqrt {a x+1}}\right )}{a} \]
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Rubi [A] time = 0.24, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {2281, 2291, 2178} \[ -\frac {F^{-\frac {n \sqrt {1-a x}}{\sqrt {a x+1}}} \left (F^{\frac {\sqrt {1-a x}}{\sqrt {a x+1}}}\right )^n \text {Ei}\left (\frac {n \sqrt {1-a x} \log (F)}{\sqrt {a x+1}}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 2178
Rule 2281
Rule 2291
Rubi steps
\begin {align*} \int \frac {\left (F^{\frac {\sqrt {1-a x}}{\sqrt {1+a x}}}\right )^n}{1-a^2 x^2} \, dx &=\left (F^{-\frac {n \sqrt {1-a x}}{\sqrt {1+a x}}} \left (F^{\frac {\sqrt {1-a x}}{\sqrt {1+a x}}}\right )^n\right ) \int \frac {F^{\frac {n \sqrt {1-a x}}{\sqrt {1+a x}}}}{1-a^2 x^2} \, dx\\ &=-\frac {\left (F^{-\frac {n \sqrt {1-a x}}{\sqrt {1+a x}}} \left (F^{\frac {\sqrt {1-a x}}{\sqrt {1+a x}}}\right )^n\right ) \operatorname {Subst}\left (\int \frac {F^{n x}}{x} \, dx,x,\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a}\\ &=-\frac {F^{-\frac {n \sqrt {1-a x}}{\sqrt {1+a x}}} \left (F^{\frac {\sqrt {1-a x}}{\sqrt {1+a x}}}\right )^n \text {Ei}\left (\frac {n \sqrt {1-a x} \log (F)}{\sqrt {1+a x}}\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 77, normalized size = 1.00 \[ -\frac {F^{-\frac {n \sqrt {1-a x}}{\sqrt {a x+1}}} \left (F^{\frac {\sqrt {1-a x}}{\sqrt {a x+1}}}\right )^n \text {Ei}\left (\frac {n \sqrt {1-a x} \log (F)}{\sqrt {a x+1}}\right )}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 25, normalized size = 0.32 \[ -\frac {{\rm Ei}\left (\frac {\sqrt {-a x + 1} n \log \relax (F)}{\sqrt {a x + 1}}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {{\left (F^{\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}}\right )}^{n}}{a^{2} x^{2} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.11, size = 0, normalized size = 0.00 \[ \int \frac {\left (F^{\frac {\sqrt {-a x +1}}{\sqrt {a x +1}}}\right )^{n}}{-a^{2} x^{2}+1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {{\left (F^{\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}}\right )}^{n}}{a^{2} x^{2} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int -\frac {{\left (F^{\frac {\sqrt {1-a\,x}}{\sqrt {a\,x+1}}}\right )}^n}{a^2\,x^2-1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {\left (F^{\frac {\sqrt {- a x + 1}}{\sqrt {a x + 1}}}\right )^{n}}{a^{2} x^{2} - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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